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Spectral/hp element methods: Recent developments, applications, and perspectives
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  • Review Article
  • Open Access
  • Published: 11 April 2018

Spectral/hp element methods: Recent developments, applications, and perspectives

  • Hui Xu1,
  • Chris D. Cantwell1,
  • Carlos Monteserin2,
  • Claes Eskilsson4,5,
  • Allan P. Engsig-Karup2,3 &
  • …
  • Spencer J. Sherwin1 

Journal of Hydrodynamics volume 30, pages 1–22 (2018)Cite this article

  • 6166 Accesses

  • 60 Citations

  • 1 Altmetric

  • Metrics details

Abstract

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C0 - continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

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Acknowledgments

The authors kindly thank the Executive Editorin-Chief Prof. Lian-di Zhou for the invitation to contribute this review article and Dr. Wei Zhang for his contribution to conducting the tip-vortex simulation. H.X. and S.J.S would like to acknowledge support under EPSRC (Grant No. EP/L000407/1).

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Authors and Affiliations

  1. Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK

    Hui Xu, Chris D. Cantwell & Spencer J. Sherwin

  2. Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Kgs., Lyngby, Denmark

    Carlos Monteserin & Allan P. Engsig-Karup

  3. Center for Energy Resources Engineering (CERE), Technical University of Denmark, 2800 Kgs., Lyngby, Denmark

    Allan P. Engsig-Karup

  4. Department of Civil Engineering, Aalborg University, DK-9220, Aalborg Ø, Denmark

    Claes Eskilsson

  5. Division Safety and Transport, Research Institutes of Sweden (RISE), SE-50115, Borås, Sweden

    Claes Eskilsson

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  1. Hui Xu
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  3. Carlos Monteserin
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Correspondence to Hui Xu.

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Xu, H., Cantwell, C.D., Monteserin, C. et al. Spectral/hp element methods: Recent developments, applications, and perspectives. J Hydrodyn 30, 1–22 (2018). https://doi.org/10.1007/s42241-018-0001-1

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  • Received: 22 December 2017

  • Accepted: 28 December 2017

  • Published: 11 April 2018

  • Issue Date: February 2018

  • DOI: https://doi.org/10.1007/s42241-018-0001-1

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Keywords

  • High-precision spectral/hp elements
  • continuous Galerkin method
  • discontinuous Galerkin method
  • implicit large eddy simulation
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